167 research outputs found
Existentially Closed Models in the Framework of Arithmetic
We prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.Ministerio de Educación y Ciencia MTM2011–2684
A saturation property of structures obtained by forcing with a compact family of random variables
A method how to construct Boolean-valued models of some fragments of
arithmetic was developed in Krajicek (2011), with the intended applications in
bounded arithmetic and proof complexity. Such a model is formed by a family of
random variables defined on a pseudo-finite sample space. We show that under a
fairly natural condition on the family (called compactness in K.(2011)) the
resulting structure has a property that is naturally interpreted as saturation
for existential types. We also give an example showing that this cannot be
extended to universal types.Comment: preprint February 201
Algebraic combinatorics in bounded induction
In this paper, new methods for analyzing models of weak subsystems of Peano Arithmetic are proposed. The focus will be on the study of algebro-combinatoric properties of certain definable cuts. Their relationship with segments that satisfy more induction, with those limited by the standard powers/roots of an element, and also with definable sets in Bounded Induction is studied. As a consequence, some considerations on the Π1-interpretability of IΔ0 in weak theories, as well as some alternative axiomatizations, are reviewed. Some of the results of the paper are obtained by immersing Bounded Induction models in its Stone-Cech Compactification, once it is endowed with a topology.Ministerio de Ciencia, Innovación y Universidades PID2019-109152GB-I0
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
Hilbert's tenth problem for weak theories of arithmetic
AbstractHilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p(x̄) from Z[x̄] whether p(x̄) has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 (bounded existential induction) or IU-1 (parameter-free bounded universal induction)
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